Emission from free particles (Chapters 18-19) T. Johnson 13-02-27 Dispersive Media, Lecture 8 - Thomas Johnson 1 Outline • Repetition: the emission formula and the multipole expansion • The emission from a free particle - the Larmor formula • Applications of the Larmor formula – Harmonic oscillator – Cyclotron radiation – Thompson scattering – Bremstrahlung • Relativistic generalisation of Larmor formula – Repetition of basic relativity – Co- and contra-variant tensor notation – Lorentz transformation and relativistic invariants – Relativistic Larmor formula • The Lienard-Wiechert potentials – Inductive and radiative electromagnetic fields – Alternative derivation of the Larmor formula • Abraham-Lorentz force 13-02-27 Dispersive Media, Lecture 8 - Thomas Johnson 2 Repetition: Emission formula • The energy emitted by a wave mode M (using antihermitian part of the propagator), when integrating over the δ-function in ω 3 WM = ∑ ∫ d kUM (k) M 2 RM (k) * UM (k) = eM (k) • Jext (ωM (k),k) ε0 – the emission formula – thus UM is a density of emission in k-space • €Alternatively, the emission per unit frequency and solid angle (dΩ) – for non-spatially dispersive media d 2Ωdω W = ∫ 3 ∑VM (ω,Ω) (2π) M * 2 ωnM eM • Jext VM (ω,Ω) = − 3 2 c * ˆ (2π ) ε0 1− eM • k 13-02-27 Dispersive Media, Lecture 8 - Thomas Johnson 3 € Repetition: The current expressed in multipole moments • Multipoles moments are related to Fourier transform of the current: J(ω,k) = ∫ dteiωt ∫ d 3x[e−ik•x ]J(t,x) ⎡ 1 2 ⎤ = dteiωt d 3x 1− ik • x − (k • x) + ... J(t,x) ∫ ∫ ⎣⎢ 2 ⎦⎥ = −iωd(ω) + ik × m(ω) − ωq • k(ω)/2 + ... Emission formula Emission spectrum (k-space power density) (integrated over solid angles) € ⎧ ⎧ 4 d RM (k) * 2 d n(ω)ω 2 ⎪U M (k) = ωM eM • d ⎪V rad (ω) = 2 3 d(ω) c ⎪ ε0 ⎪ 6π ε0 3 4 ⎪ 2 ⎪ m RM (k) * m n(ω) ω 2 ⎨U M (k) = eM • (k × m) ⎨V rad (ω) = 2 5 m(ω) ⎪ ε0 ⎪ 6π ε0c 3 6 ⎪ 2 ⎪ q RM (k) * q n(ω) ω * ⎪U M (k) = ωM eM ,iqij k j ⎪V rad (ω) = 2 5 dij dij ⎩ 4ε0 ⎩ 720π ε0c 13-02-27 Electromagnetic Processes In Dispersive Media, 4 Lecture 7 € € Repetition: Emission in the time-domain • The total radiated power from a dipole T 1 ⎛ 1 2 ⎞ 1 2 P lim dt d˙˙ (t) d˙˙ (t) ave = 3 ⎜ T → ∞ ∫ ⎟ = 3 6πε0c ⎝ 2T −T ⎠ 6πε0c – interpreted as a time average power – ideally the averaging should beperformed over all times € • for event, or periodic motion, averaging can be done over finite times 13-02-27 Dispersive Media, Lecture 8 - Thomas Johnson 5 Dipole current from single particle • Current from a single particle: J(t,x) = qX˙ ( t)δ(x − X(t)) ∞ J(ω,k) = q ∫ dte−iωt ∫ d 3k eik•x X˙ ( t)δ(x − X(t)) −∞ ∞ € = −iωq ∫ dte−iωt [1+ ik • X(t) + ...]X(t)δ(x − X(t)) −∞ ⎧ ∞ ⎫ = −iωq⎨ X(ω) + ∫ dte−iωt [ik • X(t) + ...]X(t)δ(x − X(t))⎬ ⎩ −∞ ⎭ Dipole: d=qX • Dipole approximation exp[ikx]~1 ; but what is the error? € – Assume oscillating motion: X˙ ( t) = vcos(ωt) ⇒ X ~ v/ω • for non-oscillating motion: emission of quanta ω occures on time-scale t~1/ω v n ω v v Dipole valid for k • X(t) ~ k ~ M ~ n ω € c ω M c non-relativistic motion 13-02-27 Dispersive Media, Lecture 8 - Thomas Johnson 6 € Relation between emission and acceleration • Emission from single particle; dipole current: J(ω,k) ≈ −iωqX(ω) 2 2 e* • ω2X(ω) ˆ q nM M VM ω,k = 3 2 ( ) * ˆ ε0(2πc) 1− e • k M € • Note that −ω 2 X ( ω ) = a ( ω ) is the acceleration: 2 2 e* • a(ω) ˆ q nM M VM ω,k = 3 2 € ( ) * ε0(2πc) 1 e kˆ € − M • – Thus emission from free particle is a response to acceleration! •€ Power radiated for isotropic transverse waves: truncated outside [-T,T] 2 2 1 T ˆ q nM (ω) 1 T Prad (ω) = lim dΩ VM ω,k = 2 3 lim a (ω) T → ∞ ∫ ∑ ( ) T → ∞ 2T M 12π ε0c 2T 13-02-27 Dispersive Media, Lecture 8 - Thomas Johnson 7 € The Larmor fomula • The emission integrated over all frequencies is related to time integral of the emitted power – in vacuume (nM=1) ∞ ∞ T 1 2 2π 2 2π 2 P ∝ lim aT (ω) dω = lim aT (t) dt = lim a(t) dt T → ∞ ∫ T → ∞ ∫ T → ∞ ∫ 2T −∞ 4T −∞ T −T The Larmor formula 2 q 2 P = a(t) € 3 6πε0c – the Larmor formula related the averaged radiated power with the average acceleration • The time average€ has the same interpretation as for dipole and is sometimes written 2 q 2 P(t) = 3 a(t) 6πε0c – but should always be interpreted as an average! 13-02-27 Dispersive Media, Lecture 8 - Thomas Johnson 8 € Applications: Harmonic oscillator • As a first example, consider the emission from a particle performing an harmonic oscillation – harmonic oscillations in one dimesion X ˙˙ 2 X(t) = x0 cos(ω0t) , a(t) = X (t) = −x0ω0 cos(ω0t) – Larmor formula: the emitted power associated with this acceleration 2 4 2 q ω0 x0 2 € P(t) = 3 cos (ω0t) 6πε0c 2 – oscillation cos (ω0t) above is not physical; average over a period € 2 4 2 q ω0 x0 P = 3 12πε0c € 13-02-27 Dispersive Media, Lecture 8 - Thomas Johnson 9 Applications: Harmonic oscillator – frequency spectum • Express the particle as a dipole d, use truncation for Fourier transform d(ω) = qF{X(t)} = πqx 0[δ(ω − ω0) + δ(ω + ω0 )] = 1 ⎡ ⎛ (ω − ω )T ⎞ ⎛ (ω + ω )T ⎞ ⎤ lim Tqx sinc 0 sinc 0 = T → ∞ 0 ⎢ ⎜ ⎟ + ⎜ ⎟ ⎥ 2 ⎣ ⎝ 2 ⎠ ⎝ 2 ⎠ ⎦ • The time-averaged power emitted from a dipole 2 1 1 ωd(ω) € P( ) lim V ( , )d lim ω = T → ∞ ∫∫ ω Ω Ω = T → ∞ 2 3 = T T 6π ε0c 2 1 1 1 ⎡ ⎛ (ω − ω )T ⎞ ⎛ (ω + ω )T ⎞ ⎤ lim Tqx sinc 0 sinc 0 = T → ∞ 2 3 ω 0 ⎢ ⎜ ⎟ + ⎜ ⎟ ⎥ = T 6π ε0c 2 ⎣ ⎝ 2 ⎠ ⎝ 2 ⎠ ⎦ q2ω 4 x 2 {see Ch. 4.5}... 0 0 = = 3 [δ(ω − ω0 ) + δ(ω + ω0)] 12πε0c 13-02-27 Dispersive Media, Lecture 8 - Thomas Johnson 10 € Applications: cyclotron emission • An important emission process from magnetised particles is from the acceleration involved in cyclotron motion – consider a charged particle moving in a static magnetic field B=Bzez q qB a(t) = v˙ (t) = v × B = −Ωe × v , Ω = m z m ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ v˙ x 0 Ω 0 vx vx (t) v⊥sin(Ωt) x(t) ρL cos(Ωt) ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ˙ ⎢v y ⎥ = ⎢ −Ω 0 0⎥ ⎢v y ⎥ ⇒ ⎢v y (t)⎥ = ⎢v ⊥cos(Ωt)⎥ ⇒ ⎢ y(t)⎥ = ⎢ ρL sin(Ωt)⎥ ⎢ ⎥ ⎢ ⎥ ⎣ v˙ z ⎦ ⎣⎢ 0 0 0⎦⎥ ⎣ vz ⎦ ⎣⎢v x (t)⎦⎥ ⎣⎢ v|| ⎦⎥ ⎣⎢ z(t)⎦⎥ ⎣⎢ v||t ⎦⎥ – where we have the Larmor radius ρL = v⊥/Ω – This describe circular motion around magnetic field lines of force called € the Larmor- or cyclotron-motion 4 2 2 q B v⊥ – The period averaged radiation: P = 3 2 € 12πε0c m – Magnetized plasma; power depends on the temperature: 2 P ∝ v⊥ ∝ T • Electron cyclotron emission is one of the most common ways to measure the temperature of a fusion plasma! 13-02-27 Dispersive€ Media, Lecture 8 - Thomas Johnson 11 € Applications: wave scattering • Consider a particle being accelerated by an external wave field E(t,x) = E0 cos(ω0t − k0 • x) ⇒ a(t) = qE(t,x)/m • The Larmor formula then tell us the average emitted power 2 q4 E € 0 P = 2 3 12πε0m c – Note: that this is only valid in vacuum (restriction of Larmor formula) • Rewrite in term of the wave energy density W0 2 2 2 € – in vacuum : W0 = ε0 E0 /4 + ε0 k × E0 /cω /4 = ε0 E0 /2 2 8π ⎛ q2 ⎞ P = ⎜ 2 ⎟ cW0 3 ⎝ 4πε0mc ⎠ € – Shown in the next page: This describe the fraction of the power density that is scattered by the particle, i.e. first absorbed and then re-emitted € 13-02-27 Dispersive Media, Lecture 8 - Thomas Johnson 12 Applications: wave scattering • The scattering process can be interpreted as a collision – Consider a density of wave quanta representing the power density W0 – The wave quanta, photons, move with velocity c (speed of light) – Imagine a charged particle as a ball with a cross section σT – Then the power scattered per unit time is given by P = σΤcW0 ω Cross section area σT • The effective cross section for of the particle wave scattering on electrons € r 8π q2 0 σ = r2 , r = Τ 3 0 0 4πε mc 2 0 Density of Photons hitting incoming this area are photons scattered € 13-02-27 Dispersive Media, Lecture 8 - Thomas Johnson 13 Applications: Thomson scattering • Scattering of waves against electrons is called Thomson scattering – from this process the classical radius of the electron was defined as 2 e −15 rele = 2 ≈ 2.8179 ×10 [m] 4πε0mec – Note: this is an effective radius for Thomson scattering and not a measure of the “real” size of the electron – in general quantum mechanics is needed to understand the behavior of € electrons at such short length scales • Examples of Thomson scattering: – In fusion devices, Thomson scattering of a high-intensity laser beam is used for measuring the electron temperatures and densities. – The continous spectrum from the solar corona is the result of the Thomson scattering of solar radiation with free electrons – The cosmic microwave background is thought to be linearly polarized as a result of Thomson scattering 13-02-27 Dispersive Media, Lecture 8 - Thomas Johnson 14 Applications: Bremsstrahlung • Bremsstrahlung (~Braking radiation) come from the acceleration associated with electrostatic collisions between charged particles (called Coulomb collisions) • Note that the electrostatic force is “long range”, E~1/r2 – thus electrostatic collisions between charged particles is a smooth continuous processes • Derivation: an electron moving near an ion with charge Ze – since the ion is heavier than the electron, we assume Xion(t)=0 – the equation of motion for the electron and the emitted power are 3 Ze2X(t) 2Z 2 ⎛ e2 ⎞ 1 m X˙˙ ( t) = − ⇒ P(t) = ⎜ ⎟ e 3 3m2c 3 4 4 4πε0 X(t) e ⎝ πε0 ⎠ X(t) – this is the Bremsstrahlung radiation at one time of one single collision • to estimate the total power from a medium we need to integrate over both the entire collision and all ongoing collisions! € 13-02-27 Dispersive Media, Lecture 8 - Thomas Johnson 15 Bremsstrahlung: Coulomb collisions • Lets try and integrate the emission over all times 3 ∞ 2Z 2 ⎛ e2 ⎞ ∞ dr W = P(t)dt